where \(\operatorname{sd}u\)  and \(\operatorname{cd}u\)  are the functions
given in (REF ). From definitions [1]}, [2]}, we know
that \(0\le \mathrm {sn}^2u\le 1\) ,
\(0\le \mathrm {cn}^2u\le 1\)  and \(0< \mathrm {dn}^2u< 1\) . Further, the relations [1]}
\(&&\begin{array}{l}\mathrm {cd}\;u=\mathrm {sn}(u+K)\quad \mbox{and}\quad (1-k^2)\;\mathrm {sd}^2u=1-\mathrm {cd}^2u,\end{array}\vspace{8.53581pt}\\ &&\mbox{where }K=K(k)=\int _0^{\frac{\pi }{2}}\left[1-k^2 \sin ^2\theta \right]^{-\frac{1}{2}}d\theta ,\nonumber \) 
